A052553 → (θ+1)²

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence

\(\left(-x - n + 7\,n x - n x^{2} + n^{2} - 9\,n^{2} x + 3\,n^{2} x^{2} + 3\,n^{3} x - 3\,n^{3} x^{2} + n^{4} x^{2}\right)\,P_{n}(x) = \left(-x + 8\,x^{2} - n - 9\,n x - n x^{2} - 4\,n x^{3} + n^{2} - n^{2} x - 19\,n^{2} x^{2} - 6\,n^{2} x^{3} + 3\,n^{3} x + 7\,n^{3} x^{2} + n^{4} x^{2} + 2\,n^{4} x^{3}\right)\,P_{n-1}(x) + \left(x^{2} + 4\,x^{3} - n x - 4\,n x^{2} - n x^{3} - 4\,n x^{4} - 3\,n^{2} x^{2} - 14\,n^{2} x^{3} - 8\,n^{2} x^{4} - n^{3} x^{3} - 4\,n^{3} x^{4}\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
1	1				1
2	1				1
3	1	4			5
4	1	8			9
5	1	12	9		22
6	1	16	27		44
7	1	20	54	16	91
8	1	24	90	64	179

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-ed2256bce5038d1b · revision 1 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-ec9d723a8ce3d936

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half affine: slope 1, intercept -1)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1;
   2: 1;
   3: 1,  4;
   4: 1,  8;
   5: 1, 12,   9;
   6: 1, 16,  27;
   7: 1, 20,  54,  16;
   8: 1, 24,  90,  64;
   9: 1, 28, 135, 160,  25;
  10: 1, 32, 189, 320, 125;

%F Formula

The row polynomials satisfy (n^2-n+(3*n^3-9*n^2+7*n-1)*x+(n^4-3*n^3+3*n^2-n)*x^2)*P(n,x) = (n^2-n+(3*n^3-n^2-9*n-1)*x+(n^4+7*n^3-19*n^2-n+8)*x^2+(2*n^4-6*n^2-4*n)*x^3)*P(n-1,x) - (n*x+(3*n^2+4*n-1)*x^2+(n^3+14*n^2+n-4)*x^3+(4*n^3+8*n^2+4*n)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 - n + (3*n^3 - n^2 - 9*n - 1)*x + (n^4 + 7*n^3 - 19*n^2 - n + 8)*x^2 + (2*n^4 - 6*n^2 - 4*n)*x^3)*P[n - 1, x] - (n*x + (3*n^2 + 4*n - 1)*x^2 + (n^3 + 14*n^2 + n - 4)*x^3 + (4*n^3 + 8*n^2 + 4*n)*x^4)*D[P[n - 1, x], x])/(n^2 - n + (3*n^3 - 9*n^2 + 7*n - 1)*x + (n^4 - 3*n^3 + 3*n^2 - n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((n^2 - n + (3*n^3 - n^2 - 9*n - 1)*x + (n^4 + 7*n^3 - 19*n^2 - n + 8)*x^2 + (2*n^4 - 6*n^2 - 4*n)*x^3)*v[i - 1] - (n*x + (3*n^2 + 4*n - 1)*x^2 + (n^3 + 14*n^2 + n - 4)*x^3 + (4*n^3 + 8*n^2 + 4*n)*x^4)*deriv(v[i - 1], x))/(n^2 - n + (3*n^3 - 9*n^2 + 7*n - 1)*x + (n^4 - 3*n^3 + 3*n^2 - n)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-34d5cc3a968e30b1
Coefficients of one sign. Every nonzero row, n >= 1: the coefficients of P_n have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).

derived from the cited steps by a stated theorem: e-34d5cc3a968e30b1
Hadamard kernel (Malo). Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)
Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.

derived from the cited steps by a stated theorem: e-34d5cc3a968e30b1

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

preserver_image euler_polynomial, shift 0 e-34d5cc3a968e30b1

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
  - gamma_polynomial scale 1, shift 0 e-01e2cd957653d3da
    
    Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?
    
    premise (sequence): Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    
    theorem_check liu_wang_derivative_lag e-b176436450506e78 root: theorem check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?
    
    preserver_image reversal, shift 0 e-73f1cbc14f4fad08
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check liu_wang_derivative_lag e-b176436450506e78 (expanded above)
    
    basis_map A008288 e-796ce5140a0b4b04
    
    Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?
    
    premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in [0, 0]
    
    no recorded property covers this requirement
    
    basis_map A008288 e-b8e0a0fa3ba7bf57
    
    Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?
    
    premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in [0, 0]
    
    no recorded property covers this requirement
    
    ↵ basis_map A008288 e-796ce5140a0b4b04 (expanded above)
    
    ↵ basis_map A008288 e-b8e0a0fa3ba7bf57 (expanded above)
  - preserver_image reversal, shift 0 e-515982525d76b9fb
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    
    preserver_image reversal, shift -1 e-24564bef34b2c2e5
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one vers…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    parent_provers paper_theorem e-0801036cdfff24d2 root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    parent_provers paper_theorem e-0d3631d0f359c9ce root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    gamma_polynomial scale 1, shift 0 e-a039cba7823b0b25
    
    Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?
    
    premise (sequence): delannoy
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    theorem_check liu_wang_derivative_lag e-15c233e4385971d1 root: theorem check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?
    
    preserver_image reversal, shift 0 e-266d85ca03a298fa
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): delannoy
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check liu_wang_derivative_lag e-15c233e4385971d1 (expanded above)
    
    basis_map A008288 e-e771e35dc9ee3fbd
    
    Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?
    
    premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
    
    no recorded property covers this requirement
    
    basis_map A008288 e-33dba7e79fe9caa8
    
    Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?
    
    premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
    
    no recorded property covers this requirement
    
    ↵ basis_map A008288 e-e771e35dc9ee3fbd (expanded above)
    
    ↵ basis_map A008288 e-33dba7e79fe9caa8 (expanded above)
    
    basis_map A007318 e-fcd19baf8f835cf0
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 1]
    
    preserver_image affine, shift 0 e-1365676e8cb0fd13
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-5, -1]
    
    theorem_check ma_wang_interval e-32a5b77539bbfdf5 root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?
    
    preserver_image affine, shift 0 e-3c4e27dedb9c54a8
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4, 0]
    
    theorem_check ma_wang_interval e-0178a3d1cd265fb6 root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?
    
    basis_map A026729 e-fa68694d112454ed
    
    Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?
    
    premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in [0, 0]
    
    no recorded property covers this requirement
    
    ↵ theorem_check ma_wang_interval e-0178a3d1cd265fb6 (expanded above)
  - parent_provers paper_theorem e-cf6c2292b090b00e root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (2)

Equal up to content, sign, reversal and a small index shift.

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.